Question

1. Consider a perfectly competitive market in good x consisting of 250 consumers with utility function:

                                      u(x,y) = xy

Denote P_x to be the price for good x and suppose P_y = 1. Each consumer has income equal to 10. There are 100 firms producing good x according to the cost function c(x) = x² + 1.

1. Derive the demand curve for good x for a constant in the market
2. Derive the market demand curve for good x
3. Derive the individual firm’s supply curve for good x
4. Derive the market supply for good x
5. Determine the equilibrium price and quantity in the market for good x
6. Is the market currently in long-run equilibrium? Why or why not?

Answer 1

U(X,Y) = xy

MUx = y

MUy = x

MRS = MUx/MU_Y = y/x

At optimal choice

MRS = Px/Py  

y/x = Px/1  

y/x = Px

y = xPx

Budget constraint  

P_xx + P_yy = I  

P_xx +  y = 10 (i)

Put y = xPx in equation (i)

P_xx + xPx = 10

2xPx = 10

x = 10/2P_x

x = 5/P_x

demand curve of good x is x = 5/P_x

b)  

since there are 250 consumers so market demand curve is

X = 250x  

X = 250(5/P_x )

X = 1250/P_x

Xd =  1250/P_x

c)  

c(x) = x² + 1  

MC = 2x  

supply curve of a singal or individual firm for good x is given by Px = MCx

Px = 2x

x = Px/2  

d)  

since there are 100 firms so market supply curve for good x

X = 100x  

X = 100(Px/2)  

X = 50Px

Xs = 50Px

e)  

Xd = Xs (in equilibrium)

1250/Px = 50Px  

1250/50 = P²x  

P²x = 25  

Px = 5  

X = 50(5) = 250  

equilibrium price Px = 5

equilibruim quantity X = 250